Olympiad problems

2013 Irish Math Olympiad, 8

Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satised by at least two pairs of integers $(x, y)$ with $1 < x \le y$.

2014 Turkey MO (2nd round), 3

Tags: inequalities , geometric inequality , geometry

Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$ through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation \[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \] holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.

2008 Thailand Mathematical Olympiad, 8

Tags: perfect square , number theory

Prove that $2551 \cdot 543^n -2008\cdot 7^n$ is never a perfect square, where $n$ varies over the set of positive integers

2014 Greece JBMO TST, 4

Tags: set theory , subset , partition , sum , set , combinatorics

Givan the set $S = \{1,2,3,....,n\}$. We want to partition the set $S$ into three subsets $A,B,C$ disjoint (to each other) with $A\cup B\cup C=S$ , such that the sums of their elements $S_{A} S_{B} S_{C}$ to be equal .Examine if this is possible when: a) $n=2014$ b) $n=2015 $ c) $n=2018$

2021 Ukraine National Mathematical Olympiad, 5

Tags: number theory , sum

Find all sets of $n\ge 2$ consecutive integers $\{a+1,a+2,...,a+n\}$ where $a\in Z$, in which one of the numbers is equal to the sum of all the others. (Bogdan Rublev)

1997 Korea - Final Round, 5

Tags: inequalities

For positive numbers $ a_1,a_2,\dots,a_n$, we define \[ A\equal{}\frac{a_1\plus{}a_2\plus{}\cdots\plus{}a_n}{n}, \quad G\equal{}\sqrt[n]{a_1\cdots a_n}, \quad H\equal{}\frac{n}{a_1^{\minus{}1}\plus{}\cdots\plus{}a_n^{\minus{}1}}\] Prove that (i) $ \frac{A}{H}\leq \minus{}1\plus{}2\left(\frac{A}{G}\right)^n$, for n even (ii) $ \frac{A}{H}\leq \minus{}\frac{n\minus{}2}{n}\plus{}\frac{2(n\minus{}1)}{n}\left(\frac{A}{G}\right)^n$, for $ n$ odd

2017 Mathematical Talent Reward Programme, MCQ: P 6

Tags: algebra , polynomial

Let $p(x)$ be a polynomial of degree 4 with leading coefficients 1. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$, $p(4)=4$. Then $p(5)=$ [list=1] [*] 5 [*] $\frac{25}{6}$ [*] 29 [*] 35 [/list]

2023 Stars of Mathematics, 3

Tags: geometry , fixed point

Let $ABC$ be an acute triangle, with $AB<AC{}$ and let $D$ be a variable point on the side $AB{}$. The parallel to $D{}$ through $BC{}$ crosses $AC{}$ at $E{}$. The perpendicular bisector of $DE{}$ crosses $BC{}$ at $F{}$. The circles $(BDF)$ and $(CEF)$ cross again at $K{}$. Prove that the line $FK{}$ passes through a fixed point. [i]Proposed by Ana Boiangiu[/i]

2024 Euler Olympiad, Round 1, 10

Tags: equation , euler , algebra

Find all $x$ that satisfy the following equation: \[ \sqrt {1 + \frac {20}x } = \sqrt {1 + 24x} + 2 \] [i]Proposed by Andria Gvaramia, Georgia [/i]

2018 Mathematical Talent Reward Programme, SAQ: P 2

Tags: polynomial , algebra

$P(x)$ is polynomial with real coefficients such that $\forall n \in \mathbb{Z}, P(n) \in \mathbb{Z}$. Prove that every coefficients of $P(x)$ is rational numbers.

2014 Costa Rica - Final Round, 5

Tags: algebra , functional equation , functional , recurrence relation

Let $f : N\to N$ such that $$f(1) = 0\,\, , \,\,f(3n) = 2f(n) + 2\,\, , \,\,f(3n-1) = 2f(n) + 1\,\, , \,\,f(3n-2) = 2f(n).$$ Determine the smallest value of $n$ so that $f (n) = 2014.$

2010 Germany Team Selection Test, 1

Tags: polynomial , algebra

Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$

2014 Kyiv Mathematical Festival, 1a

Tags:

a) 2 white and 2 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 4, to the other black cat is 8. The sum of distances from the black cats to one white cat is 3, to the other white cat is 9. What cats are sitting on the edges? b) 2 white and 3 black cats are sitting on the line. The sum of distances from the white cats to one black cat is 11, to another black cat is 7 and to the third black cat is 9. The sum of distances from the black cats to one white cat is 12, to the other white cat is 15. What cats are sitting on the edges? [size=85](Kyiv mathematical festival 2014)[/size]

2002 India Regional Mathematical Olympiad, 3

Tags: adfghj

Let $a,b,c$ be positive integers such that $a$ divides $b^2$, $b$ divides $c^2$ and $c$ divides $a^2$. Prove that $abc$ divides $(a + b +c)^7$.

2019 Polish MO Finals, 1

Tags: concyclic , orthocenter , tangent , phantom point , geometry

Let $ABC$ be an acute triangle. Points $X$ and $Y$ lie on the segments $AB$ and $AC$, respectively, such that $AX=AY$ and the segment $XY$ passes through the orthocenter of the triangle $ABC$. Lines tangent to the circumcircle of the triangle $AXY$ at points $X$ and $Y$ intersect at point $P$. Prove that points $A, B, C, P$ are concyclic.

1976 Bundeswettbewerb Mathematik, 4

Tags: geometry , 3d geometry , sphere , geometric transformation , vector , combinatorics

Each vertex of the 3-dimensional Euclidean space either is coloured red or blue. Prove that within those squares being possible in this space with edge length 1 there is at least one square either with three red vertices or four blue vertices !

2010 Korea - Final Round, 5

Tags: induction , symmetry , combinatorics

On a circular table are sitting $ 2n$ people, equally spaced in between. $ m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule. (i) One may give cookies only to people adjacent to himself. (ii) In order to give a cookie to one's neighbor, one must eat a cookie. Select arbitrarily a person $ A$ sitting on the table. Find the minimum value $ m$ such that there is a strategy in which $ A$ can eventually receive a cookie, independent of the distribution of cookies at the beginning.

Dumbest FE I ever created, 4.

Tags: function , algebra

Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$ for all real number $x$ and $y$

1988 China Team Selection Test, 1

Tags: number theory

Let $f(x) = 3x + 2.$ Prove that there exists $m \in \mathbb{N}$ such that $f^{100}(m)$ is divisible by $1988$.

2016 PUMaC Team, 15

Tags: number theory

Compute the sum of all positive integers $n$ with the property that $x^n \equiv 1$ (mod $2016$) has $n$ solutions in $\{0, 1, 2, ... , 2015\}$.

2018 Polish Junior MO Finals, 5

Tags: geometry , equilateral triangle , plane geometry

Point $M$ is middle of side $AB$ of equilateral triangle $ABC$. Points $D$ and $E$ lie on segments $AC$ and $BC$, respectively and $\angle DME = 60 ^{\circ}$. Prove that, $AD + BE = DE + \frac{1}{2}AB$.

Durer Math Competition CD 1st Round - geometry, 2018.C+2

Tags: geometry , isosceles , right triangle , angle

In an isosceles right-angled triangle $ABC$, the right angle is at $A$. $D$ lies so on the side $BC$ that $2CD = DB$. Let $E$ be the projection of $B$ onto $AD$. What is the measure fof angle $\angle CED $?

2011 Albania Team Selection Test, 3

Tags: ratio , hyperbola , trigonometry , analytic geometry , circumcircle , geometry , conic

In the acute angle triangle $ABC$ the point $O$ is the center of the circumscribed circle and the lines $OA,OB,OC$ intersect sides $BC,CA,AB$ respectively in points $M,N,P$ such that $\angle NMP=90^o$. [b](a)[/b] Find the ratios $\frac{\angle AMN}{\angle NMC}$,$\frac{\angle AMP}{\angle PMB}$. [b](b)[/b] If any of the angles of the triangle $ABC$ is $60^o$, find the two other angles.

2020 Thailand TSTST, 2

Tags: inequalities

Let $x, y, z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that $$\frac{x+1}{z+x+1}+\frac{y+1}{x+y+1}+\frac{z+1}{y+z+1}\geq\frac{(xy+yz+zx+\sqrt{xyz})^2}{(x+y)(y+z)(z+x)}.$$

2012 BMT Spring, 4

Tags: combinatorics

There are 1$2$ people labeled $1, ..., 12$ working together on $12$ missions, with people $1, ... , i $working on the $i$th mission. There is exactly one spy among them. If the spy is not working on a mission, it will be a huge success, but if the spy is working on the mission, it will fail with probability $1/2$. Given that the first $11$ missions succeed, and the $12$th mission fails, what is the probability that person $12$ is the spy?